Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.
Source: Peru EGMO TST 2021 #5
Tags: number theory
Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.